KK Monopoles As Giant Gravitons in Ads5

KK Monopoles as Giant Gravitons in AdS S5 5 ×

Bert Janssen Universidad de Granada & CAFPE

In collaboration with: Y. Lozano (Oviedo) and D. Rodr´ıguez Gomez´ (Princeton) References: in preparation

B. Janssen (UGR) 1/24 Outlook

1. Introduction: Giant gravitons as p-branes • Giant gravitons as gravitational waves •

B. Janssen (UGR) 2/24 Outlook

1. Introduction: Giant gravitons as p-branes • Giant gravitons as gravitational waves • 2. A new giant graviton in AdS S5 5 × Gravitational waves expanding into S5 •

B. Janssen (UGR) 2/24 Outlook

1. Introduction: Giant gravitons as p-branes • Giant gravitons as gravitational waves • 2. A new giant graviton in AdS S5 5 × Gravitational waves expanding into S5 • 3. KK-monopoles as giant gravitons Effective actions for KK- monopoles • The giant graviton solution • Generalisation to Sasaki-Einstein Manifolds •

B. Janssen (UGR) 2/24 Outlook

B. Janssen (UGR) 2/24 1 Introduction to giant gravitons

1.1 Giant gravitons as p-branes

n (n 2) (n 2)-brane in AdS S , wrapped around S − and moving in φ: − m × 2 2 2 2 2 2 2 2 ds = dsAdS + L dθ + cos θdφ + sin θdΩ(n 2) −

2 Pφ 2 N˜ n 3 E = L 1 + tan θ 1 P sin − θ L sin θ − φ r ϕ θ Minima: sin θ = 0: Pointlike graviton P 1 L φ n−3 sin θ = ( N˜ ) : Giant graviton

Pφ Energy: E(minimum) = L [McGreevy, Susskind, Toumbas]

Realisation of Stringy Exclusion Principle −→ B. Janssen (UGR) 3/24 1.2 Giant gravitons as gravitational waves

Abelian picture: (n 2)-brane with momentum in worldvolume − Non-Abelian picture: Gravitational waves expanding into fuzzy (n 2)-brane − due to dielectric effect [McGreevy, Susskind, Toumbas] [BMN] ...

B. Janssen (UGR) 4/24 1.2 Giant gravitons as gravitational waves

1 1 i kj S = T dτ STr k− P [E + E (Q δ) E E ] det Q − W − 00 0i − − k j0 Z q 2 (1) (3) 1 2 (6) + T dτ STr P [k− k ] + iP [(i i )C ] + P [(i i ) i C ] + W − X X 2 X X k · · · Z 2 1 (3) i i i k E = g k− k k + k− (i C ) , Q = δ + ik[X , X ]E µν µν − µ ν k µν j j kj

[B.J., Lozano]

Abelian wave action for X i 1l • ∼ Myers’ action for multiple D0-branes after dim reduction • First order: Matrix string theory in inﬁnite momentum frame •

B. Janssen (UGR) 4/24 1.2 Giant gravitons as gravitational waves Abelian picture: (n 2)-brane with momentum in worldvolume − Non-Abelian picture: Gravitational waves expanding into fuzzy (n 2)-brane − due to dielectric effect [McGreevy, Susskind, Toumbas] [BMN] ...

1 1 i kj S = TW dτ STr k− P [E + E i(Q δ) E Ej ] det Q − − 00 0 − − k 0 Z q 2 (1) (3) 1 2 (6) + TW dτ STr P [k− k ] + iP [(iX iX )C ] + P [(iX iX ) ikC ] + − 2 · · · Z 2 1 (3) i i i k E = g k− k k + k− (i C ) , Q = δ + ik[X , X ]E µν µν − µ ν k µν j j kj [B.J., Lozano] Abelian wave action for X i 1l • ∼ Myers’ action for multiple D0-branes after dim reduction • First order: Matrix string theory in inﬁnite momentum frame • Gauged sigma model in kµ (= propagation direction of waves): • @ dynamical embedding scalar for this direction

B. Janssen (UGR) 4/24 AdS S4: W M2 as fuzzy S2 7 × −→

i L sin θ i i j ijk k X = 2 J [J , J ] = 2i J √N 1 , with − = XiXi = L2 sin2 θ1l ⇒ i P

B. Janssen (UGR) 5/24 AdS S4: W M2 as fuzzy S2 7 × −→

i L sin θ i i j ijk k X = 2 J [J , J ] = 2i J √N 1 , with − = XiXi = L2 sin2 θ1l ⇒ i P

T 4L sin θ 4L4 sin2 θ cos2 θ 4L2 sin2 θ E = W STr 1l X2 + X2 + X2X2 L cos θ s − √N 2 1 N 2 1 N 2 1   − − − 

 3 2  NTW 2 2L 5 = 1 + tan θ 1 sin θ + (N − ) L s − √N 2 1 O − P 2L3 2 E = φ 1 + tan2 θ 1 sin θ L s − N h i

Agreement for large N, upon identiﬁcation Pφ = NTW −→ [B.J., Lozano]

B. Janssen (UGR) 5/24 AdS S5: W D3 as fuzzy S3 = U(1) ﬁbre bundle over fuzzy S2 5 × −→ Type IIB wave action: extra isometry due to T-duality

2 2 1 1 Φ (4) Eµν = gµν k− kµkν `− `µ`ν k− `− e (iki`C )µν, − − − i i i k Φ Qj = δj + i[X , X ]e− k`Ekj.

Identify isometry direction with ﬁbre direction of S3 −→ [B.J., Lozano, Rodr´ıguez-Gomez]´

B. Janssen (UGR) 6/24 AdS S5: W D3 as fuzzy S3 = U(1) ﬁbre bundle over fuzzy S2 5 × −→ Type IIB wave action: extra isometry due to T-duality

2 2 1 1 Φ (4) E = g k− k k `− ` ` k− `− e (i i C ) , µν µν − µ ν − µ ν − k ` µν i i i k Φ Qj = δj + i[X , X ]e− kÈkj. Identify isometry direction with fibre direction of S3 −→ [B.J., Lozano, Rodr´ıguez-Gomez]´ AdS S7: W M5 as fuzzy S5 = U(1) fibre bundle over fuzzy CP 2 4 × −→ Technical reasons: Add momentum in fibre direction

5 (6) 1 2 (1) Macroscopically: S = T d ξ P [ikC ] P [k− k ] F F − 5 − − 2 ∧ ∧ R n o 2 with 2 F F = 8π N CP ∧ R

2 (1) 2 (1) = S T5 dτP [k− k ] F F = NT0 dτP [k− k ] ⇒ ∼ − 2 ∧ Z ZCP Z [B.J., Lozano, Rodr´ıguez-Gomez]´

B. Janssen (UGR) 6/24 2 A new giant graviton in AdS S5 5 × 2 2 2 5 r 2 r 1 2 r 2 2 ds = (1 + )dt + (1 + )− dr + dΩ + (dχ + A) − L2 L2 4 2 h i 2 2 2 +L dΣ4 + (dψ + B) h i with 1 A = cos χ dχ B = sin2 ϕ (dϕ + cos ϕ dϕ ) 1 2 2 1 4 2 3

1 2 dΣ2 = dϕ2 + sin2 ϕ dϕ2 + sin2 ϕ dϕ2 + cos2 ϕ dϕ + cos ϕ dϕ 4 1 4 1 2 2 3 1 4 2 3

B. Janssen (UGR) 7/24 2 A new giant graviton in AdS S5 5 × 2 2 2 5 r 2 r 1 2 r 2 2 ds = (1 + )dt + (1 + )− dr + dΩ + (dχ + A) − L2 L2 4 2 h i 2 2 2 +L dΣ4 + (dψ + B) h i with 1 A = cos χ dχ B = sin2 ϕ (dϕ + cos ϕ dϕ ) 1 2 2 1 4 2 3

1 2 dΣ2 = dϕ2 + sin2 ϕ dϕ2 + sin2 ϕ dϕ2 + cos2 ϕ dϕ + cos ϕ dϕ 4 1 4 1 2 2 3 1 4 2 3 χ and ψ global isometry directions • Type IIB wave action exhibits two isometries • (propagation direction & T-duality)

waves expand into S5 via fuzzy CP 2? −→

B. Janssen (UGR) 7/24 1 1 i jk S = T dτ STr k− P E E (Q δ) E E det Q W − W − µν − µi − − j kν Z n r h i o 2 (1) (4) 1 2 (7) + TW dτ STr P [k− k ] iP [(iX iX )i`C ] P [(iX iX ) i`ikN ] + . . . − − − 2 Z n o where 1 1 (4) 2 2 E = k− `− (i i C ) = g k− k k `− ` ` µν Gµν − k ` µν Gµν µν − µ ν − µ ν µ µ µ ρ ρ ν µ Q ν = δν + ik` [X , X ]Eρν ((iX iX )i`C4)λ = [X , X ]` Cµνρλ

ρ ν (i`ikΩ)µ1...µn = ` k Ωνρµ1...µn

double gauged sigma model: • @ dynamical embedding scalars for kµ and `µ

B. Janssen (UGR) 8/24 1 1 i jk S = T dτ STr k− P E E (Q δ) E E det Q W − W − µν − µi − − j kν Z n r h i o 2 (1) (4) 1 2 (7) + TW dτ STr P [k− k ] iP [(iX iX )i`C ] P [(iX iX ) i`ikN ] + . . . − − − 2 Z n o where 1 1 (4) 2 2 E = k− `− (i i C ) = g k− k k `− ` ` µν Gµν − k ` µν Gµν µν − µ ν − µ ν µ µ µ ρ ρ ν µ Q ν = δν + ik` [X , X ]Eρν ((iX iX )i`C4)λ = [X , X ]` Cµνρλ

ρ ν (i`ikΩ)µ1...µn = ` k Ωνρµ1...µn

double gauged sigma model: • @ dynamical embedding scalars for kµ and `µ

N (7) gauge ﬁeld of Kaluza-Klein monopole with `µ as Taub-NUT • (7) (1) d(N ) = ∗d(` )

ρ ν (i`ikΩ)µ1...µn = ` k Ωνρµ1...µn

double gauged sigma model: • @ dynamical embedding scalars for kµ and `µ

N (7) gauge ﬁeld of Kaluza-Klein monopole with `µ as Taub-NUT • (7) (1) d(N ) = ∗d(` )

Choose: kµ = δµ `µ = δµ Xi fuzzy CP 2 • ψ χ3 → B. Janssen (UGR) 8/24 CP 2 is coset manifold SU(3)/U(2) or embedded in R8 via

8 8 1 xixi = 1 dijkxjxk = xi √ i=1 j,k 3 X X=1 Fuzzy CP 2 generated by SU(3) generators J i in (anti-)fundamental repres

J i if ijk Xi = [Xi, Xj] = Xk (2n2 2)/3 (2n2 2)/3 − − p p[Alexanian, Balachandran, Immirzi, Ydri]

B. Janssen (UGR) 9/24 CP 2 is coset manifold SU(3)/U(2) or embedded in R8 via

8 8 1 xixi = 1 dijkxjxk = xi √ i=1 j,k 3 X X=1 Fuzzy CP 2 generated by SU(3) generators J i in (anti-)fundamental repres

J i if ijk Xi = [Xi, Xj] = Xk (2n2 2)/3 (2n2 2)/3 − − p p[Alexanian, Balachandran, Immirzi, Ydri]

2 6 2 2 1 r 3L r 2 S = TW dτ STr L− 1 + 1l + X − s L2 32(N 1) Z n h − i o 2 6 2 NTW r L r 5 E(r) = 1 + 1 + + (N − ) L L2 32(N 1) O r h − i

B. Janssen (UGR) 9/24 CP 2 is coset manifold SU(3)/U(2) or embedded in R8 via

8 8 1 xixi = 1 dijkxjxk = xi √ i=1 j,k 3 X X=1 Fuzzy CP 2 generated by SU(3) generators J i in (anti-)fundamental repres J i if ijk Xi = [Xi, Xj] = Xk (2n2 2)/3 (2n2 2)/3 − − p p[Alexanian, Balachandran, Immirzi, Ydri]

2 6 2 2 1 r 3L r 2 S = TW dτ STr L− 1 + 1l + X − s L2 32(N 1) Z n h − i o 2 6 2 NTW r L r 5 E(r) = 1 + 1 + + (N − ) L L2 32(N 1) O r h − i

NT P = H(0) = W = ψ (topological) giant graviton! ⇒ L L

B. Janssen (UGR) 9/24 Fuzzy brane with ﬁnite extension and massless dispersion relation • (topological) giant graviton −→

B. Janssen (UGR) 10/24 Fuzzy brane with ﬁnite extension and massless dispersion relation • (topological) giant graviton −→

Check supersymmetry... •

B. Janssen (UGR) 10/24 Fuzzy brane with ﬁnite extension and massless dispersion relation • (topological) giant graviton −→

Check supersymmetry... •

What is macroscopical interpretation? • (NB: inverse dielectric problem)

B. Janssen (UGR) 10/24 Fuzzy brane with ﬁnite extension and massless dispersion relation • (topological) giant graviton −→

Check supersymmetry... •

What is macroscopical interpretation? • (NB: inverse dielectric problem)

S 1 T dτ STr P [(i i )2i i N (7)] W ∼ − 2 W X X ` k Z n o Fuzzy CP 2: (i i )2 = 0 X X 6 KK-monopole with Taub-NUT in `µ and wrapped on kµ −→

B. Janssen (UGR) 10/24 Fuzzy brane with ﬁnite extension and massless dispersion relation • (topological) giant graviton −→

Check supersymmetry... •

What is macroscopical interpretation? • (NB: inverse dielectric problem)

S 1 T dτ STr P [(i i )2i i N (7)] W ∼ − 2 W X X ` k Z n o Fuzzy CP 2: (i i )2 = 0 X X 6 KK-monopole with Taub-NUT in `µ and wrapped on kµ −→

What is ﬁeld theory dual on CFT side? •

B. Janssen (UGR) 10/24 3 KK monopoles as giant gravitons 3.1 Effective actions for KK monopoles

Kaluza-Klein monopole:

2 a b 1 i 2 i j ds = ηabdx dx + H− (dz + Aidy ) + Hδijdy dy

i with ∂i∂ H = 0 2∂[iAj] = ijk∂kH [Sorkin] [Gross, Perry]

B. Janssen (UGR) 11/24 3 KK monopoles as giant gravitons 3.1 Effective actions for KK monopoles

Kaluza-Klein monopole:

2 a b 1 i 2 i j ds = ηabdx dx + H− (dz + Aidy ) + Hδijdy dy

i with ∂i∂ H = 0 2∂[iAj] = ijk∂kH [Sorkin] [Gross, Perry] Effective action: Taub-NUT direction z is isometry: not dynamical embedding scalar! gauged sigma model: −→ µ µ 2 ν µ µ µ X = ∂ X `− ` ∂ X ` with ` = δ Da a − ν a z effective metric: −→ µ µ 2 µ ν P [g] = g X X = g `− ` ` ∂ X ∂ X µν Da Db µν − µ ν a b [Bergshoeff, B.J., Ort´ın]

B. Janssen (UGR) 11/24 Intermezzo: Gauged sigma models Eliminate embedding scalars in presence of isometry no dymanical degree of freedom −→

E.g. Effective actions for gravitational wave from T-duality

S = T d2σ det(∂ Xµ∂ Xνg ) F 1 − 1 | α β µν | Z q Orientate strings as X µ(τ, σ) = (Xa(τ), X9(σ)) = (Xa(τ), mσ)

B. Janssen (UGR) 12/24 Intermezzo: Gauged sigma models Eliminate embedding scalars in presence of isometry no dymanical degree of freedom −→

E.g. Effective actions for gravitational wave from T-duality

S = T d2σ det(∂ Xµ∂ Xνg ) F 1 − 1 | α β µν | Z q Orientate strings as X µ(τ, σ) = (Xa(τ), X9(σ)) = (Xa(τ), mσ) Apply T-duality along X 9 wave propagating along X 9 −→

1 µ ν 2 a S = mT dτ k− ∂X ∂X + k− k ∂X W − 0 | Gµν | a Z n q o 2 with: = g k− k k Gµν µν − µ ν µ µ 9 k = δ9 is Killing vector along X k = g , k2 = k = g −→ µ µx x xx

B. Janssen (UGR) 12/24 X9 is no dymanical degree of freedom (since Killing)

µ µ µ 2 k = k g k k− k k Gµν µν − µ ν

= kν kν = 0 −

∂Xµ∂Xν has only contributions for X µ = X9 Gµν 6 Wave action S has only contributions for X µ = X9 6

B. Janssen (UGR) 13/24 X9 is no dymanical degree of freedom (since Killing)

µ µ µ 2 k = k g k k− k k Gµν µν − µ ν

= kν kν = 0 − ∂Xµ∂Xν has only contributions for X µ = X9 Gµν 6 Wave action S has only contributions for X µ = X9 6

Equivalently:

1 µ ν 9 S = mT dτ k− X X g + A ∂X . W − 0 |D D µν | − Z n q o µ µ µ µ 2 ρ µ with X ∂X Ak ∂X k− k ∂X k D ≡ − ≡ − ρ SW invariant under

δXµ = Λ(τ)kµ δA = ∂Λ(τ)

X9 is pure gauge, not a dymanical degree of freedom −→ B. Janssen (UGR) 13/24 Type IIB KK monopole: (2, 0) tensor multiplet in 6 dim ﬁeld content: 3 embedding scalars X i −→ 2 worldvolume scalars ω, ω˜ + selfdual 2-form Wab

B. Janssen (UGR) 14/24 Type IIB KK monopole: (2, 0) tensor multiplet in 6 dim ﬁeld content: 3 embedding scalars X i −→ 2 worldvolume scalars ω, ω˜ + selfdual 2-form Wab

KK wrapped around S5 in AdS S5 with Taub-NUT in `µ = δµ 5 × χ U(1) ﬁbre direction in KK worldvolume compactify over kµ = δµ −→ ψ [Eyras, B.J., Lozano]

B. Janssen (UGR) 14/24 Type IIB KK monopole: (2, 0) tensor multiplet in 6 dim ﬁeld content: 3 embedding scalars X i −→ 2 worldvolume scalars ω, ω˜ + selfdual 2-form Wab

KK wrapped around S5 in AdS S5 with Taub-NUT in `µ = δµ 5 × χ U(1) ﬁbre direction in KK worldvolume compactify over kµ = δµ −→ ψ [Eyras, B.J., Lozano] wrapped KK monopole: (1,1) vector multiplet in 5-dim ﬁeld content: 3 embedding scalars X i −→ 2 worldvolume scalars ω, ω˜ ( truncate here) −→ 1 vector Va double gauged sigma model:

µ µ 2 ν µ 2 ν µ X = ∂ X `− ` ∂ X ` k− k ∂ X k Da a − ν a − ν a effective worldvolume: R CP 2 −→ × B. Janssen (UGR) 14/24 3.2 The giant graviton solution

5 2φ 2 φ 1 1 S = T d σ e− k` det(P [g] + e k ` ) − 4 | ab − − Fab | Z p 1 kµ T d5σ P [i i N (7)] P + . . . − 4 ` k − 2 k2 ∧ F ∧ F Z n h i o where µ µ 2 2 µ ν P [g] = g X X = g `− ` ` k− k k ∂ X ∂ X µν Da Db µν − µ ν − µ ν a b = 2∂V (1) + P [i i C(4)] F k `

B. Janssen (UGR) 15/24 3.2 The giant graviton solution

Now choose:

wrapped KK with worldvolume CP 2 • = √2n dB where B is S5 ﬁbre connection • F

B. Janssen (UGR) 15/24 3.2 The giant graviton solution

5 2φ 2 φ 1 1 S = T d σ e− k` det(P [g] + e k ` ) − 4 | ab − − Fab | Z p 1 kµ T d5σ P [i i N (7)] P + . . . − 4 ` k − 2 k2 ∧ F ∧ F Z n h i o where µ µ 2 2 µ ν P [g] = g X X = g `− ` ` k− k k ∂ X ∂ X µν Da Db µν − µ ν − µ ν a b = 2∂V (1) + P [i i C(4)] F k ` Now choose: wrapped KK with worldvolume CP 2 • = √2n dB where B is S5 ﬁbre connection • F ? dB = (dB), dB dB √g 2 √g 5 ∧ ∼ CP ∼ S = = 8π2n2 : non-trivial instanton number on CP 2 ⇒ CP 2 F ∧ F Z [Trautman]

B. Janssen (UGR) 15/24 Chern-Simons:

T4 2 µ 2 2 µ P k− k F F = n TW dt P k− k 2 R CP 2 ∧ ∧ Z × h i Z h i n2 units of momentum in ψ dissolved in worldvolume (4π2T = T ) −→ 4 0

B. Janssen (UGR) 16/24 Chern-Simons:

T4 2 µ 2 2 µ P k− k F F = n TW dt P k− k 2 R CP 2 ∧ ∧ Z × h i Z h i n2 units of momentum in ψ dissolved in worldvolume (4π2T = T ) −→ 4 0

Born-Infeld: det(g + ) is perfect square if = √2ndB αβ Fαβ F Lr2 r2 32n2 2 S = T dt dΣ 1 + L4 + g 4 4 4 L2 L2r2 | Σ| Z r h i 2 2 6 2 8n T4 r L r = dt 1 + 2 2 + 1 dΣ4 gΣ − L r L 32n | | Z h i Z p

B. Janssen (UGR) 16/24 Chern-Simons:

T4 2 µ 2 2 µ P k− k F F = n TW dt P k− k 2 R CP 2 ∧ ∧ Z × h i Z h i n2 units of momentum in ψ dissolved in worldvolume (4π2T = T ) −→ 4 0

n2T r2 L6r2 = E(r) = 0 1 + 1 + ⇒ L L2 32n2 r h i 2 6 2 NTW r L r 5 E(r) = 1 + 1 + + (N − ) L L2 32(N 1) O r h − i B. Janssen (UGR) 16/24 3.3 Generalization to Sasaki-Einstein manifolds

Y is Sasaki-Einstein manifold = (Y ) is Kahler¨ & Ricci-ﬂat • ⇐⇒ M C 2 2 2 2 with ds = dr + r dsY M

B. Janssen (UGR) 17/24 3.3 Generalization to Sasaki-Einstein manifolds

Y is Sasaki-Einstein manifold = (Y ) is Kahler¨ & Ricci-ﬂat • ⇐⇒ M C 2 2 2 2 with ds = dr + r dsY M Y is U(1) bundle over Kahler¨ -Einstein manifold M • 5 4 2 2 2 dsY = dΣ4 + (dψ + B)

2 with: dΣ4 metric on M4 µ µ k = δψ is Killing vector (Reeb vector) 1 ωM = 2 dB is Kahler¨ form on M4 M has topology S2 S2 4 ×

B. Janssen (UGR) 17/24 3.3 Generalization to Sasaki-Einstein manifolds

Y is Sasaki-Einstein manifold = (Y ) is Kahler¨ & Ricci-ﬂat • ⇐⇒ M C 2 2 2 2 with ds = dr + r dsY M Y is U(1) bundle over Kahler¨ -Einstein manifold M • 5 4 2 2 2 dsY = dΣ4 + (dψ + B)

2 with: dΣ4 metric on M4 µ µ k = δψ is Killing vector (Reeb vector) 1 ωM = 2 dB is Kahler¨ form on M4 M has topology S2 S2 4 × S1 S1 E.g.: R6 = (S5 CP 2) Conifold= (S3 S2 S2 S2) • C −→ C × −→ ×

B. Janssen (UGR) 17/24 3.3 Generalization to Sasaki-Einstein manifolds

Y is Sasaki-Einstein manifold = (Y ) is Kahler¨ & Ricci-ﬂat • ⇐⇒ M C 2 2 2 2 with ds = dr + r dsY M Y is U(1) bundle over Kahler¨ -Einstein manifold M • 5 4 2 2 2 dsY = dΣ4 + (dψ + B)

2 with: dΣ4 metric on M4 µ µ k = δψ is Killing vector (Reeb vector) 1 ωM = 2 dB is Kahler¨ form on M4 M has topology S2 S2 4 × S1 S1 E.g.: R6 = (S5 CP 2) Conifold= (S3 S2 S2 S2) • C −→ C × −→ × AdS/CFT conjecture: • strings in AdS Y dual to = 1 CFT on boundary of AdS 5 × 5 N

B. Janssen (UGR) 17/24 AdS Y : 5 × 5 2 2 2 5 r 2 r 1 2 r 2 2 ds = (1 + )dt + (1 + )− dr + dΩ + (dχ + A) − L2 L2 4 2 h i 2 2 2 +L dΣ4 + (dψ + B) h i

B. Janssen (UGR) 18/24 AdS Y : 5 × 5 2 2 2 5 r 2 r 1 2 r 2 2 ds = (1 + )dt + (1 + )− dr + dΩ + (dχ + A) − L2 L2 4 2 h i 2 2 2 +L dΣ4 + (dψ + B) h i Wrapped monopole with

`µ = δµ in Taub-NUT • χ kµ = δµ as extra isometry • ψ such that • F = 8π2n2 F ∧ F ZM NB: always possible thanks to S2 S2 topology of M × 4 NB: for S5 and S3 S2: = √2ndB (also in general?) × F

B. Janssen (UGR) 18/24 AdS Y : 5 × 5 2 2 2 5 r 2 r 1 2 r 2 2 ds = (1 + )dt + (1 + )− dr + dΩ + (dχ + A) − L2 L2 4 2 h i 2 2 2 +L dΣ4 + (dψ + B) h i Wrapped monopole with

= det(g + ) is perfect square ⇒ F

B. Janssen (UGR) 18/24 5 2φ 2 φ 1 1 S = T d σ e− k` det(P [g] + e k ` ) − 4 | ab − − Fab | Z p r2 8n2 2 = T dt dΣ k`2 1 + L4 + g 4 4 L2 k2`2 | Σ| Z r h i 8n2Ω T r2 L4k2`2 = Σ 4 dt 1 + + 1 − k2 L2 8n2 Z r h i

B. Janssen (UGR) 19/24 5 2φ 2 φ 1 1 S = T d σ e− k` det(P [g] + e k ` ) − 4 | ab − − Fab | Z p r2 8n2 2 = T dt dΣ k`2 1 + L4 + g 4 4 L2 k2`2 | Σ| Z r h i 8n2Ω T r2 L4k2`2 = Σ 4 dt 1 + + 1 − k2 L2 8n2 Z r h i

n2T r2 L6r2 = E(r) 0 1 + 1 + ⇒ ∼ L L2 32n2 r h i giant graviton at r = 0 −→

Microscopically: Gravitational waves expanding into fuzzy M4...

B. Janssen (UGR) 19/24 4 A ﬁeld theory interpretation?

S3 ﬁbre χ is Taub-NUT direction of KK-monopole AdS in global coordinates −→ 5 S1 = 4 SU(K) SYM on S3 S2 −→ N −→

B. Janssen (UGR) 20/24 4 A ﬁeld theory interpretation?

S3 ﬁbre χ is Taub-NUT direction of KK-monopole AdS in global coordinates −→ 5 S1 = 4 SU(K) SYM on S3 S2 −→ N −→ = 4 scalar action: N µ 1 1 2 S dt dΩ Tr ∂ Φ∗∂ Φ + Φ∗Φ + [Φ , Φ∗] ∼ 3 µ a a L2 a a 4g2 a b Z n o with Φa in adjoint repres of SU(K) in fundamental repres of SU(3) (explicit R-symmetry)

B. Janssen (UGR) 20/24 4 A ﬁeld theory interpretation?

2 2 1 1 2 S dt dΩ Tr ∂ Φ∗∂ Φ 4Φ∗ (S )Φ + Φ∗Φ + [Φ , Φ∗] ∼ 2 − t a t a − a∇ a L2 a a 4g2 a b Z n o B. Janssen (UGR) 20/24 Ansatz: Φ = eif(t) J a ⊗ Ma ⊗ a with: f(t) arbitrary function of time

2 Ja generators of SU(2) : constant on S

a va va a = diag(v , K 1 , ... , K 1 ): SU(K) gauge dependence M − − − −

B. Janssen (UGR) 21/24 Ansatz: Φ = eif(t) J a ⊗ Ma ⊗ a with: f(t) arbitrary function of time

2 Ja generators of SU(2) : constant on S

a va va a = diag(v , K 1 , ... , K 1 ): SU(K) gauge dependence M − − − −

Action: 2 1 2 2 v S = α dt v (f 0) + 2 2L2 Z h i

B. Janssen (UGR) 21/24 Ansatz: Φ = eif(t) J a ⊗ Ma ⊗ a with: f(t) arbitrary function of time

2 Ja generators of SU(2) : constant on S

a va va a = diag(v , K 1 , ... , K 1 ): SU(K) gauge dependence M − − − −

Action: 2 1 2 2 v S = α dt v (f 0) + 2 2L2 Z h i Hamiltonian: 2 2 p v ∂S 2 0 H = 2 + 2 with p = = v f 2v 2L ∂f 0

2 Minimum for v0 = pL p = H(v2) = Massless particle! ⇒ 0 L

B. Janssen (UGR) 21/24 = 4 scalar action with SU(3) R-symmtry N

µ 1 1 2 S dt dΩ Tr ∂µΦ∗∂ Φa + Φ∗Φa + [Φa, Φ∗] ∼ 3 a L2 a 4g2 b Z n o Ansatz Φ = eif(t) J SU(2) symmetry a ⊗ Ma ⊗ a

B. Janssen (UGR) 22/24 = 4 scalar action with SU(3) R-symmtry N

µ 1 1 2 S dt dΩ Tr ∂µΦ∗∂ Φa + Φ∗Φa + [Φa, Φ∗] ∼ 3 a L2 a 4g2 b Z n o Ansatz Φ = eif(t) J SU(2) symmetry a ⊗ Ma ⊗ a

= R-remaining symmetry SU(3)/U(2) CP 2 ⇒ ' = worldvolume of wrapped KK monopole

B. Janssen (UGR) 22/24 = 4 scalar action with SU(3) R-symmtry N

µ 1 1 2 S dt dΩ Tr ∂µΦ∗∂ Φa + Φ∗Φa + [Φa, Φ∗] ∼ 3 a L2 a 4g2 b Z n o Ansatz Φ = eif(t) J SU(2) symmetry a ⊗ Ma ⊗ a

= R-remaining symmetry SU(3)/U(2) CP 2 ⇒ ' = worldvolume of wrapped KK monopole

Generalisation to Sasaki-Einstein: S1 = 1 SCFT with gauge group in S3 S2 • N G −→

B. Janssen (UGR) 22/24 = 4 scalar action with SU(3) R-symmtry N

µ 1 1 2 S dt dΩ Tr ∂ Φ∗∂ Φ + Φ∗Φ + [Φ , Φ∗] ∼ 3 µ a a L2 a a 4g2 a b Z n o Ansatz Φ = eif(t) J SU(2) symmetry a ⊗ Ma ⊗ a

= R-remaining symmetry SU(3)/U(2) CP 2 ⇒ ' = worldvolume of wrapped KK monopole

Generalisation to Sasaki-Einstein: S1 = 1 SCFT with gauge group in S3 S2 • N G −→ Scalar sector with R-symmetry group G • R Φ independent of ﬁbre and constant in S2 • a Φ = eif(t) J a ⊗ Ma ⊗ a

Remaining R-symmetry GR/U(2) M = worldvol of wrapped KK • ' 4 B. Janssen (UGR) 22/24 Explicit example: AdS S3 S2 5 × ×

Gravity side: wrapped KK-monopoles on S2 S2 × Gauge theory side: scalars with SU(2) SU(2) U(1) R-symmetry × × Ansatz remaining symmetry −→ SU(2) SU(2) U(1) SU(2) SU(2) × × × U(2) ∼ SU(2)

diagonal SU(2): same winding number over each S2! −→

B. Janssen (UGR) 23/24 5 Conclusions

Multiple gravitational waves in AdS S5 expanding into S5: • 5 × new, topological giant graviton −→

B. Janssen (UGR) 24/24 5 Conclusions

Multiple gravitational waves in AdS S5 expanding into S5: • 5 × new, topological giant graviton −→

Macroscopically: KK monopole wrapped on S5 with Taub-NUT in AdS • 5

B. Janssen (UGR) 24/24 5 Conclusions

Multiple gravitational waves in AdS S5 expanding into S5: • 5 × new, topological giant graviton −→

Macroscopically: KK monopole wrapped on S5 with Taub-NUT in AdS • 5

Tecnically: wrapped monopole, compactiﬁed over S5 ﬁbre • 2 µ 2 add momentum by P [k− k ] with = 8πn −→ ∧ F ∧ F F ∧ F R R

B. Janssen (UGR) 24/24 5 Conclusions

Multiple gravitational waves in AdS S5 expanding into S5: • 5 × new, topological giant graviton −→

Macroscopically: KK monopole wrapped on S5 with Taub-NUT in AdS • 5

Tecnically: wrapped monopole, compactiﬁed over S5 ﬁbre • 2 µ 2 add momentum by P [k− k ] with = 8πn −→ ∧ F ∧ F F ∧ F R R if(t) Field Theory: scalars Φa = e a Ja • ⊗ M ⊗ preserve SU(3)/U(2) CP 2 −→ '

Easily generalisable to AdS Y with Y Sasaki-Einstein • 5 × 5 5

B. Janssen (UGR) 24/24